How to define indeterminates formally?

Question

Indeterminate are used in algebra to define polynomials and formal power series. Here is the definition of polynomials:

A polynomial in an indeterminate X is an expression of the form $ a_0 + a_1X + a_2X^2 + \ldots + a_nX^n,$ where the $a_i$ are called the coefficients of the polynomial.

In the definition above there are two vague terms, namely "indeterminate " and "expression ". My understanding of classical math is that we should define every mathematical object and definition in terms of the language of $ZFC$.

My question is this: how does one define indeterminates and expressions by sets?

Answer 1

You can define a polynomial over a ring $R$ as a map $a:\mathbb N\to R$ with $\{n\mid a(n)\neq0\}$ finite.

Then you define multiplication and addition of polynomials. $$(a+b)(n)=a(n)+b(n)\\(a\cdot b)(n) = \sum_{k=0}^n a(k)b(n-k)$$

Show that this forms a ring.

Then the "indeterminate" is $x(1)=1$ and $x(n)=0$ if $n\neq 1$.

(It's a little more difficult to define what "x" is when $R$ is a ring without identity. Then, it is just a place-holder, almost.)

Then you'll find that $$a = \sum_{k: a(k)\neq 0} a(k)x^k.$$

The formal power series ring is a little more complicated. You define it essentially the same way, without the finiteness condition, but you can't technically say $a=\sum_{k=0}^\infty a_kz^k$ in that case. There is an way around this as defining the ring $R[[x]]$ as the inverse limit of $R[x]/x^n$. In that sense, $R[[x]]$ is like the $p$-adic numbers, and you can actually talk about limits.

A common powerful concept is the concept of a "universal property." The ring of polynomials $R[x]$, when $R$ is a commutative, has a nice universal property:

(1) There is a homomorphism $R\to R[x]$ and an element $x\in R[x]$.

(2) Given any ring $S$, a homomorphism $R\to S$ and an $s\in S$, then there is exactly one homomorphism $R[x]\to S$ with $x\mapsto s$.

These conditions are enough to define $R[x]$ up to isomorphism. But note that there are a lot of automorphism of $R[x]$.

Answer 2

From Iain T. Adamson's Introduction to Field Theory, p.26:

In elementary books on algebra, polynomials are usually defined to be "expressions of the form $$f(x) ≡ a_0 + a_1x + a_2x^2 + … + a_nx^n$$ where $a_0$, …, $a_n$ are numbers". This definition is open to at least two objections. First, it gives no indication as to the logical status of the symbol $x$—and to say that $x$ is a "variable" or an "indeterminate" simply begs further questions. Secondly, when one comes to define addition and multiplication of polynomials, it is difficult to avoid the feeling that these operations are at least partially defined already—for the polynomials are written with addition signs between their terms, and these terms contain powers of $x$. To avoid these objections we proceed in what may appear to be a more abstract way; in reality we are simply exploiting the well-known fact that in elementary algebra the powers of $x$ act essentially only as "place-holders" while the coefficients are the really important constituents. These remarks may already have suggested to the sophisticated reader that we might define polynomials to be simply finite sequences of coefficients such as $(a_0, a_1, …, a_n)$. For technical reasons—in fact, to enable us to deal conveniently with polynomials of different degrees, which would correspond to sequences of different lengths—we prefer to deal with "essentially finite" infinite sequences. We now proceed with the formal development.

Let $R$ be a commutative ring with identity element $e$. We denote by $P(R)$ the set of infinite sequences $(a_0, a_1, …, a_n, …)$ of elements of $R$, each of which has the property that only finitely many of the members $a_i$ of the sequence are non-zero; thus for each sequence $a = (a_0, a_1, a_2, …)$ in $P(R)$ there is an integer $N_a$ such that $a_i = 0$ for all integers $i > N_a$. It is important to be clear that two sequences are equal if and only if corresponding members are equal, i.e. if $a = (a_0, a_1, a_2, …)$ and $b = (b_0, b_1, b_2, …)$ then $a = b$ if and only if $a_i = b_i (i = 0, 1, 2, …)$.

We introduce an operation of addition in $P(R)$ by setting $$(a_0, a_1, a_2, …)+(b_0, b_1, b_2, …) = (a_0 + b_0, a_1 + b_1, a_2 + b_2, …).$$ It is easily verified that under this law of composition $P(R)$ forms an abelian group. The zero element is clearly the sequence $z = (0, 0, 0, …)$ each of whose members is the zero of $R$; the additive inverse of $(a_0, a_1, a_2, …)$ is $(-a_0, -a_1, -a_2, …)$.

Next we introduce an operation of multiplication in $P(R)$ by setting $$(a_0, a_1, a_2, …)(b_0, b_1, b_2, …) = (c_0, c_1, c_2, …),$$ where $$c_n = \sum_{i=0}^n a_i b_{n−i},\quad (n = 0, 1, 2, …).$$ Then it is a routine matter to verify that this multiplication is associative and commutative, and also distributive with respect to the addition. That is to say, $P(R)$ is a commutative ring under the laws of composition we have defined. Further, $P(R)$ has an identity element, namely the sequence $(e, 0, 0, …)$ where $e$ is the identity element of $R$.

Consider now the mapping $κ$ of $R$ into $P(R)$ defined by setting $κ(a_0) = (a_0, 0, 0, …)$ for all elements $a_0$ of $R$. It is easy to see that $κ$ is a monomorphism; we call it the canonical monomorphism of $R$ into $P(R)$. Then $R$ and its image $κ(R)$ under $κ$ are isomorphic; they differ, of course, in the nature of their elements, but have exactly the same structure. We frequently find it convenient to blur the distinction between $R$ and $κ(R)$ and to use the same symbol $a_0$ for both an element of $R$ and for its image under $κ$ in $P(R)$; when we do this, we say that we are identifying $R$ with its image under $κ$ and regarding $R$ as a subring of the ring $P(R)$. It will be found in practice that very little confusion is likely to arise from this identification procedure; but any confusion which does arise can be resolved by a return to the strictly logical notation.

We now introduce a name for the special sequence $(0, e, 0, 0, …)$ in $P(R)$: we call it $X$. By induction we can prove at once that, for every positive integer $n$, $X^n$ is the sequence $(c_0, c_1, c_2, …)$ for which $c_n = e$ and $c_i = 0$ whenever $i \ne n$. Then if $f = (a_0, a_1, …, a_N, 0, 0, …)$ is any sequence in $P(R)$, with $a_n = 0$ for all integers $n > N$, we have $$ f = (a_0, 0, 0, …) + (0, a_1, 0, …) + … + (0, …, 0, a_N, 0, …)$$ $$ = κ(a_0) + κ(a_1)X + … + κ(a_N)X^N.$$ Carrying out the identification of $R$ and $κ(R)$ described in the last paragraph, we see that we have expressed $f$ in the form $$f = a_0 + a_1X + … + a_NX^N.$$ This provides justification for calling the elements of $P(R)$ polynomials and for describing $P(R)$ as the ring of polynomials with coefficients in $R$.

Answer 3

Here's one way to define the ring of polynomials $R[x]$ without resorting to such terms (where we take $\mathbb N = \{0,1,2,\ldots\}$):

Define $R[x]$ as the set of maps $\mathbb N\rightarrow R$ with only finitely many* $n$ mapping to non-zero elements of $R$. For $f,g\in R[x]$ we define addition and multiplication as follows: $$(f+g)(n)=f(n)+g(n)$$ $$(fg)(n) = \sum_{i=0}^{n}f(i)g(n-i)$$

Essentially, for any such $f$ the value $f(n)$ represents the coefficient of $x^n$ in such a power series. Indeed, you can see that the map: $$\pi(f)=\sum_{n=0}^{\infty}f(n)x^n$$ is a homomorphism between the above definition and the intuitive algebraic one.

One can also, though with more effort, define notions of "formal sums" and accomplish it this way. To give a sketch of that, notice that we can define any expression as a tree of its components, where each node represents the application of some operation to its children. We then take the "quotient" by a certain equivalence relation, but this can be somewhat difficult to express in closed form. To get a ring of polynomials, one takes the trees built with addition, negation, and multiplication with leaves as elements of $R\cup \{x\}$ and then defines the equivalence relation $\cong$ to satisfy replacement axioms like: $$\text{If }a\cong a' \text{ and }b\cong b' \text{ then }ab\cong a'b'$$ for each operation, as well as satisfy commutative ring axioms like $$a(bc)\cong (ab)c.$$ Then, we define $\cong$ to be the "finest" or "smallest" relation satisfying this (which may be constructed by intersecting every relation satisfying all these). Then, we end up with the quotient of the trees of expressions by the relation $\cong$ being a representation of $R[x]$. This looks nasty, but it's really quite intuitive; basically, what it says is "$R[x]$ is the algebraic structure where we take $R$ and add a new element $x$, and say two expressions are equal if and only if we can prove they are equal from ring axioms" and shows that this sort of construction is well defined (which is good, because things like this show up in algebra all the time)

(*For formal power series, we remove this requirement)